Node Editor, Unwrapping, and FBX export not working
Sorry if this reply comes a little late, but I rarely ever import into Unity, so I had to do a bit of trial and error.

First of all, there is nothing wrong with your file. The only thing that is missing is the link to your texture. If you load it back in via the Texture-Tab in the Properties Window, everything works as it should. 



Consider packing your textures into the .blend files if you hand them off to somebody else, this will incorporate the images into the file and you will no longer have to send the texture files along with the .blend file.

Now, to import those files into Unity, there are a couple of workflows. I will give you the one that I find the most useful, if you want to learn about all the ins and outs, i would recommend Darrin Lile's course on Blender and Unity Pipelines. It is only about 12 Dollars, but it goes through all possible permutations of asset importing and gives you best practises to follow along the way. Like for instance naming the materials. It will not matter with a single cube, like in your file, but the materials do pile up. 

Darrin Lile's Creating a Game Environment in Unity Course

To get assets from Blender into Unity follow these steps:


Set up the materials (like you have)
Export an FBX. Either for the whole Scene, or, with the "Selected" Checkbox checked, only the selected Objects
Drag these objects into Unity's Asset Window (or right click it and select Import New Asset)





I would recommend you set up 3 Folders: Models for the Mesh Data, Materials for the Unity Materials and Textures for the Texture files





If you use the same Texture Folder for your Unity Project that you use for your .blend files, you will not have to import anything here. It populates with the files that are in this folder.
Now select your object. It has the material on it, that you assigned in Blender. You need to hook up the brick Texture manually to the Albedo slot (see image above). Drag it from the texture folder onto the square next to the Albedo Label.


This should be all you need. I remember there was an option to embed textures into an FBX, but as far as I could find out, that ONLY works if you use Blender Internal Renderer and do NOT use nodes to build the material. Just wanted to mention it for completion's sake. FBX Export Article

Also, on import, the default color in Unity is an off-white, which will make your textures look less contrasty. To counteract this, set the base color to full white.